which is a variant of the modified Friedmann equation. As for the second Friedmann (acceleration) equation, its general form was derived in [1]} for general non-Riemannian cosmological setups. It reads
\(\frac{\ddot{a}}{a}=-\frac{1}{3}R_{\mu \nu }u^{\mu }u^{\nu }+2\left( \frac{\dot{a}}{a} \right)\Phi +2\dot{\Phi }+\left( \frac{\dot{a}}{a} \right)\left(A+\frac{C}{2}\right)+\frac{\dot{A}}{2}-\frac{A^{2}}{2}-\frac{1}{2}AC-2A\Phi -2C \Phi \) 
